Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13545 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2101 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | #135 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,3)(5,6,11,15,13,20,27,33,10,16,9,26,12,7,17,19)(8,14,23,30,28,35,36,21,25,32,22,31,24,18,29,34);(1,2,4,3)(5,7,10,20)(6,13,16,12)(8,21)(9,26,11,15)(14,29)(17,33,27,19)(18,23)(22,35)(24,31)(25,34)(28,30)(32,36);(1,3,4,2)(5,8,9,22,13,28,17,29,10,25,11,23,12,24,27,36)(6,14,15,30,20,35,33,21,16,32,26,31,7,18,19,34)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=d,d^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,d^b=dh,f^b=fhi,g^b=ghi,h^b=hi,e^c=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239857, #72239861, #72240849, … |
| maximal normal subgroup of | #72239857, #72239861, #72240849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56256);